Advances in Applied Mathematics最新文献

We consider the following card guessing game with no feedback. An ordered deck of n cards labeled 1 up to n is riffle-shuffled exactly one time. Then, the goal of the game is to maximize the number of correct guesses of the cards. One after another a single card is drawn from the top, the guesser makes a guess without seeing the card and gets no response if the guess was correct or not. Building upon and improving earlier results, we provide a limit law for the number of correct guesses and also show convergence of the integer moments.

We first show that an adjoint of a loopless matroid is connected if and only if the original matroid is connected. By proving that the opposite lattice of a modular matroid is isomorphic to its extension lattice, we obtain that a modular matroid has only one adjoint (up to isomorphism) which can be given by its opposite lattice. This makes projective geometries become a key ingredient in characterizing the adjoint sequence $a{d}^{0}M,adM,a{d}^{2}M,\dots$ of a connected matroid M. We classify such adjoint sequences into three types: finite, cyclic and convergent. For the first two types, the adjoint sequences eventually stabilize at finite projective geometries except for free matroids. For the last type, the infinite non-repeating adjoint sequences are convergent to infinite projective geometries.

Recently, Dixit et al. [24] established a very elegant generalization of Hardy's theorem concerning the infinitude of zeros that the Riemann zeta function possesses at its critical line.

By introducing a general transformation formula for the theta function involving the Bessel and modified Bessel functions of the first kind, we extend their result to a class of Dirichlet series satisfying Hecke's functional equation. In the process, we also find new generalizations of classical identities in Analytic number theory.

The famous J.C.P. Miller formula provides a recurrence algorithm for the composition $B_a\circ f$, where $B_a$ is the formal binomial series and f is a formal power series, however it requires that f has to be a nonunit.

In this paper we provide the general J.C.P. Miller formula which eliminates the requirement of nonunitness of f and, instead, we establish a necessary and sufficient condition for the existence of the composition $B_a\circ f$. We also provide the general J.C.P. Miller recurrence algorithm for computing the coefficients of that composition, if $B_a\circ f$ is well defined, obviously. Our generalizations cover both the case in which f is a one–variable formal power series and the case in which f is a multivariable formal power series.

In the central part of this article we state, using some combinatorial techniques, the explicit form of the general J.C.P. Miller formula for one-variable case.

As applications of these results we provide an explicit formula for the inverses of polynomials and formal power series for which the inverses exist, obviously. We also use our results to investigation of approximate solution to a differential equation which cannot be solved in an explicit way.

We study substitutions on a countably infinite alphabet (without compactification) as Borel dynamical systems. We construct stationary and non-stationary generalized Bratteli-Vershik models for a class of such substitutions, known as left determined. In this setting of Borel dynamics, using a stationary generalized Bratteli-Vershik model, we provide a new and canonical construction of shift-invariant measures (both finite and infinite) for the associated class of subshifts.

Building off recent work of Garg and Peng, we continue the investigation into classical and consecutive pattern avoidance in rooted forests, resolving some of their conjectures and questions and proving generalizations whenever possible. Through extensions of the forest Simion-Schmidt bijection introduced by Anders and Archer, we demonstrate a new family of forest-Wilf equivalences, completing the classification of forest-Wilf equivalence classes for sets consisting of a pattern of length 3 and a pattern of length at most 5. We also find a new family of nontrivial c-forest-Wilf equivalences between single patterns using the forest analogue of the Goulden-Jackson cluster method, showing that a ${(1-o(1\left)\right)}^n$-fraction of patterns of length n satisfy a nontrivial c-forest-Wilf equivalence and that there are c-forest-Wilf equivalence classes of patterns of length n of exponential size. Additionally, we consider a forest analogue of super-strong-c-Wilf equivalence, introduced for permutations by Dwyer and Elizalde, showing that super-strong-c-forest-Wilf equivalences are trivial by enumerating linear extensions of forest cluster posets.

Ordering identities in the Weyl–Heisenberg algebra generated by single-mode boson operators are investigated. A boson string composed of creation and annihilation operators can be expanded as a linear combination of other such strings, the simplest example being a normal ordering. The case when each string contains only one annihilation operator is already combinatorially nontrivial. Two kinds of expansion are derived: (i) that of a power of a string Ω in lower powers of another string ${\Omega }^{\prime }$, and (ii) that of a power of Ω in twisted versions of the same power of ${\Omega }^{\prime }$. The expansion coefficients are shown to be, respectively, generalized Stirling numbers of Hsu and Shiue, and certain generalized Eulerian numbers. Many examples are given. These combinatorial numbers are binomial transforms of each other, and their theory is developed, emphasizing schemes for computing them: summation formulas, Graham–Knuth–Patashnik (GKP) triangular recurrences, terminating hypergeometric series, and closed-form expressions. The results on the first type of expansion subsume a number of previous results on the normal ordering of boson strings.

Much of the captivating numerology surrounding finite reflection groups stems from Solomon's celebrated 1963 theorem describing invariant differential forms. Invariant differential derivations also exhibit fascinating numerology over the complex numbers linked to rational Catalan combinatorics. We explore the analogous theory over arbitrary fields, in particular, when the characteristic of the underlying field divides the order of the acting reflection group and the conclusion of Solomon's Theorem may fail. Using results of Broer and Chuai, we give a Saito criterion (Jacobian criterion) for finding a basis of differential derivations invariant under a finite group that distinguishes certain cases over fields of characteristic 2. We show that the reflecting hyperplanes lie in a single orbit and demonstrate a duality of exponents and coexponents when the transvection root spaces of a reflection group are maximal. A set of basic derivations are used to construct a basis of invariant differential derivations with a twisted wedging in this case. We obtain explicit bases for the special linear groups $\text{SL}(n,q)$ and general linear groups $\text{GL}(n,q)$, and all groups in between.

It is well-known that the central path of semi-definite optimization, unlike linear optimization, has no analytic extension to $\mu =0$ in the absence of the strict complementarity condition. In this paper, we consider a reparametrization $\mu \mapsto {\mu }^{\rho }$, with ρ being a positive integer, that recovers the analyticity of the central path at $\mu =0$. We investigate the complexity of computing ρ using algorithmic real algebraic geometry and the theory of complex algebraic curves. We prove that the optimal ρ is bounded by $2^{O(m^2+n^{2}m+n^4)}$, where n is the matrix size and m is the number of affine constraints. Our approach leads to a symbolic algorithm, based on the Newton-Puiseux algorithm, which computes a feasible ρ using $2^{O(m+n^2)}$ arithmetic operations.